Solving Logarithmic Equations With A Graphing Calculator

Hey guys! Ever get stuck trying to solve a logarithmic equation? Don't worry, it happens to the best of us. Logarithmic equations might seem intimidating at first, but with the help of a graphing calculator, you can easily find the solutions. In this article, we'll break down the process of using a graphing calculator to solve the logarithmic equation $\log _4(8 x-6)-\log _4(x+1)=1$. We'll walk through each step, so you can tackle similar problems with confidence. Let's dive in!

Understanding Logarithmic Equations

Before we jump into using the graphing calculator, let's quickly recap what logarithmic equations are and some key properties that will help us. Logarithmic equations are equations where the variable appears inside a logarithm. Remember, a logarithm is essentially the inverse operation of exponentiation. For example, the equation $\log_b(a) = c$ is equivalent to $b^c = a$. Understanding this relationship is crucial for solving logarithmic equations.

One of the key properties we'll use in this example is the quotient rule of logarithms. This rule states that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, it's expressed as:

$\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$

This property allows us to combine the two logarithmic terms in our equation into a single logarithm, making it easier to solve. Keep this in mind as we move forward!

Step 1: Simplify the Equation Using Logarithmic Properties

Okay, let's get started with our equation: $\log _4(8 x-6)-\log _4(x+1)=1$. The first thing we want to do is simplify the equation using the quotient rule of logarithms we just talked about. Remember, the quotient rule states that $\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$. Applying this rule to our equation, we get:

$\log _4(\frac{8x-6}{x+1}) = 1$

Great! We've successfully combined the two logarithmic terms into one. This makes the equation much easier to work with. Now, we need to get rid of the logarithm to isolate the variable x. To do this, we'll use the definition of a logarithm.

Recall that $\log_b(a) = c$ is equivalent to $b^c = a$. Applying this to our simplified equation, where b is 4, a is $\frac{8x-6}{x+1}$, and c is 1, we get:

$4^1 = \frac{8x-6}{x+1}$

Which simplifies to:

$4 = \frac{8x-6}{x+1}$

Now we have a rational equation, which we can solve by multiplying both sides by $(x+1)$ to get rid of the fraction. This will lead us to a linear equation that we can easily solve.

Step 2: Convert the Logarithmic Equation to a Rational Equation

Alright, let's continue solving our equation. We left off with:

$4 = \frac{8x-6}{x+1}$

To eliminate the fraction, we'll multiply both sides of the equation by $(x+1)$. This gives us:

$4(x+1) = 8x-6$

Now, we need to distribute the 4 on the left side of the equation:

$4x + 4 = 8x - 6$

We've successfully transformed the rational equation into a linear equation. This is a significant step forward because linear equations are much easier to solve. Our next goal is to isolate the variable x on one side of the equation. We can do this by moving the terms with x to one side and the constant terms to the other side.

Step 3: Set Up Equations for the Graphing Calculator

Before we jump to the graphing calculator, let's rearrange the linear equation we obtained in the previous step. We have:

$4x + 4 = 8x - 6$

To solve this using a graphing calculator, we'll set up two separate equations, one for each side of the equation. This will allow us to find the point of intersection, which represents the solution to the original equation. Let's define our two equations as:

$y_1 = 4x + 4$

$y_2 = 8x - 6$

Now, we have two linear equations in slope-intercept form (y = mx + b). This makes it easy to input them into the graphing calculator. We'll enter $y_1$ as one function and $y_2$ as another function in the calculator. The x-coordinate of the point where these two lines intersect will be the solution to our equation.

It's important to understand why we're doing this. We're essentially looking for the value of x that makes the left side of the original equation equal to the right side. Graphically, this is represented by the intersection point of the two lines.

Step 4: Input the Equations into the Graphing Calculator

Okay, guys, let's fire up those graphing calculators! The exact steps might vary slightly depending on the model of your calculator, but the general process is the same. We're going to input the two equations we just defined into the calculator's equation editor.

First, press the "Y=" button on your calculator. This will bring up the equation editor where you can enter the functions. You'll see a list of "Y" variables, such as Y1, Y2, Y3, etc. We'll use Y1 and Y2 for our equations.

In the Y1 slot, enter the equation $4x + 4$. You'll typically use the "X,T,θ,n" button to enter the variable x. So, you'll type something like "4X + 4".

Next, in the Y2 slot, enter the equation $8x - 6$. Again, use the "X,T,θ,n" button for the x variable. You'll type "8X - 6".

Double-check that you've entered the equations correctly. A small typo can lead to a wrong solution. Once you're confident that the equations are entered correctly, we're ready to graph them and find their intersection point.

Step 5: Graph the Equations and Find the Intersection Point

Now that we have our equations entered into the graphing calculator, it's time to graph them and find their intersection point. This point represents the solution to our equation.

First, press the "GRAPH" button on your calculator. This will display the graphs of the two equations, $y_1 = 4x + 4$ and $y_2 = 8x - 6$. You should see two lines intersecting on the screen. If you don't see the intersection point clearly, you may need to adjust the window settings of your graph.

To adjust the window, press the "WINDOW" button. This will allow you to change the minimum and maximum values for the x and y axes. You can experiment with different values until you see the intersection point clearly. A standard window setting (Xmin = -10, Xmax = 10, Ymin = -10, Ymax = 10) often works well, but you might need to adjust it depending on the specific equations.

Once you can see the intersection point, we'll use the calculator's "intersect" function to find its coordinates. This function will give us the exact x and y values of the intersection point.

To access the "intersect" function, press "2nd" and then "TRACE" (which is the "CALC" button). This will bring up a menu of calculation options. Select option 5, which is "intersect".

The calculator will then prompt you to select the first curve, the second curve, and a guess for the intersection point. For each prompt, use the up and down arrow keys to select the appropriate curve (Y1 and Y2) and press "ENTER". For the guess, you can move the cursor close to the intersection point and press "ENTER".

The calculator will then display the coordinates of the intersection point. The x-coordinate of this point is the solution to our equation!

Step 6: Interpret the Solution and Check for Extraneous Solutions

Awesome! We've found the intersection point using the graphing calculator. The calculator should display the coordinates of the intersection, something like (x, y) = (2.5, 14). Remember, we're interested in the x-coordinate, as it represents the solution to our equation.

So, in this case, the solution appears to be x = 2.5. However, we're not quite done yet! It's crucial to check our solution for extraneous solutions. Extraneous solutions are solutions that we obtain algebraically but don't actually satisfy the original equation. This can happen with logarithmic equations because the domain of a logarithmic function is restricted to positive values.

To check for extraneous solutions, we'll plug our solution, x = 2.5, back into the original equation: $\log _4(8 x-6)-\log _4(x+1)=1$.

Let's substitute x = 2.5:

$\log _4(8(2.5)-6)-\log _4(2.5+1)=1$

$\log _4(20-6)-\log _4(3.5)=1$

$\log _4(14)-\log _4(3.5)=1$

Now, we can use the quotient rule of logarithms to simplify this:

$\log _4(\frac{14}{3.5})=1$

$\log _4(4)=1$

Since $\log _4(4)$ is indeed equal to 1 (because $4^1 = 4$), our solution x = 2.5 is valid and not an extraneous solution.

Therefore, the solution to the logarithmic equation $\log _4(8 x-6)-\log _4(x+1)=1$ is x = 2.5.

Conclusion

And there you have it, guys! We've successfully solved a logarithmic equation using a graphing calculator. We walked through each step, from simplifying the equation using logarithmic properties to graphing the equations and finding the intersection point. Remember to always check your solutions for extraneous roots to ensure they are valid.

Using a graphing calculator can be a powerful tool for solving logarithmic equations, especially those that are difficult or impossible to solve algebraically. By understanding the underlying principles and following these steps, you can confidently tackle a wide range of logarithmic equations. Keep practicing, and you'll become a pro at solving these types of problems!